A304998 -id:A304998 - cp's OEIS frontend

A002494 Number of n-node graphs without isolated nodes.

1, 0, 1, 2, 7, 23, 122, 888, 11302, 262322, 11730500, 1006992696, 164072174728, 50336940195360, 29003653625867536, 31397431814147073280, 63969589218557753586160, 245871863137828405125824848, 1787331789281458167615194471072, 24636021675399858912682459613241920

Offset: 0

N. J. A. Sloane

Number of unlabeled simple graphs covering n vertices. - Gus Wiseman, Aug 02 2018

			From _Gus Wiseman_, Aug 02 2018: (Start)
Non-isomorphic representatives of the a(4) = 7 graphs:
  (12)(34)
  (12)(13)(14)
  (12)(13)(24)
  (12)(13)(14)(23)
  (12)(13)(24)(34)
  (12)(13)(14)(23)(24)
  (12)(13)(14)(23)(24)(34)
(End)

F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
W. L. Kocay, Some new methods in reconstruction theory, Combinatorial Mathematics IX, 952 (1982) 89--114. [From Benoit Jubin, Sep 06 2008]
W. L. Kocay, On reconstructing spanning subgraphs, Ars Combinatoria, 11 (1981) 301--313. [From Benoit Jubin, Sep 06 2008]
J. H. Redfield, The theory of group-reduced distributions, Amer. J. Math., 49 (1927), 433-435; reprinted in P. A. MacMahon, Coll. Papers I, pp. 805-827.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..75 (using A000088)
P. C. Fishburn and W. V. Gehrlein, Niche numbers, J. Graph Theory, 16 (1992), 131-139.
R. J. Mathar, Illustrations (2023)
J. H. Redfield, The theory of group-reduced distributions [Annotated scan of pages 452 and 453 only]
Eric Weisstein's World of Mathematics, Isolated Point.
Eric Weisstein's World of Mathematics, Graphical Partition
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Equals first differences of A000088. Cf. A006129 (labeled), A001349 (connected, inv. Euler Transf).
Cf. also A006647, A006648, A006649, A006650, A006651.
Cf. A000612, A001187, A055621, A304998.

b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
      +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
       add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
    end:
a:= n-> b(n$2, [])-`if`(n>0, b(n-1$2, []), 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2019

<< MathWorld`Graphs`
Length /@ (gp = Select[ #, GraphicalPartitionQ] & /@
Graphs /@ Range[9])
nn = 20; g = Sum[NumberOfGraphs[n] x^n, {n, 0, nn}]; CoefficientList[Series[ g (1 - x), {x, 0, nn}], x]  (*Geoffrey Critzer, Apr 14 2012*)
sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]==2&]],Union@@#==Range[n]&]]],{n,6}] (* Gus Wiseman, Aug 02 2018 *)
b[n_, i_, l_] := If[n==0 || i==1, 1/n!*2^(Function[p, Sum[Ceiling[(p[[j]]-1)/2] + Sum[GCD[p[[k]], p[[j]]], {k, 1, j-1}], {j, 1, Length[p]}]][Join[l, Table[1, {n}]]]), Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]];
a[n_] := b[n, n, {}] - If[n > 0, b[n-1, n-1, {}], 0];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 03 2019, after Alois P. Heinz *)

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A002494(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q,r in p.items()),prod(q**r*factorial(r) for q,r in p.items())) for p in partitions(n))-sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q,r in p.items()),prod(q**r*factorial(r) for q,r in p.items())) for p in partitions(n-1))) if n else 1 # Chai Wah Wu, Jul 03 2024

O.g.f.: (1-x)*G(x) where G(x) is o.g.f. for A000088. - Geoffrey Critzer, Apr 14 2012

a(n) = A327075(n)+A001349(n). - R. J. Mathar, Nov 21 2023

More terms from Vladeta Jovovic, Apr 10 2000

a(0) added from David W. Wilson, Aug 24 2008

A305854 Number of unlabeled spanning intersecting set-systems on n vertices.

Original entry on oeis.org

1, 1, 2, 10, 110, 14868, 1289830592

Offset: 0

Gus Wiseman, Jun 11 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge.

			Non-isomorphic representatives of the a(3) = 10 spanning intersecting set-systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{3},{1,3},{2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A261006, A283877, A304998, A305843, A305844, A305855-A305857.

a(n) = A305856(n) - A305856(n-1) for n > 0. - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

a(6) from Bert Dobbelaere, Apr 28 2025

A261005 Number of unlabeled simplicial complexes with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210, 789204635842035039135545297410259322

Offset: 0

N. J. A. Sloane, Aug 13 2015

nonn

Also the number of non-isomorphic antichains of nonempty sets covering n vertices. The labeled case is A006126, except with a(0) = 1. - Gus Wiseman, Feb 23 2019

			From _Gus Wiseman_, Feb 23 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(4) = 20 antichains:
  {}  {{1}}  {{12}}    {{123}}         {{1234}}
             {{1}{2}}  {{1}{23}}       {{1}{234}}
                       {{13}{23}}      {{12}{34}}
                       {{1}{2}{3}}     {{14}{234}}
                       {{12}{13}{23}}  {{1}{2}{34}}
                                       {{134}{234}}
                                       {{1}{24}{34}}
                                       {{1}{2}{3}{4}}
                                       {{13}{24}{34}}
                                       {{14}{24}{34}}
                                       {{13}{14}{234}}
                                       {{12}{134}{234}}
                                       {{1}{23}{24}{34}}
                                       {{124}{134}{234}}
                                       {{12}{13}{24}{34}}
                                       {{14}{23}{24}{34}}
                                       {{12}{13}{14}{234}}
                                       {{123}{124}{134}{234}}
                                       {{13}{14}{23}{24}{34}}
                                       {{12}{13}{14}{23}{24}{34}}
(End)

Benoît Jubin, Posting to Sequence Fans Mailing List, Aug 12 2015.

C. Lienkaemper, A. Shiu, and Z. Woodstock, Obstructions to convexity in neural codes, Preprint 2015.
Francisco Ponce Carrión and Seth Sullivant, Marginal Independence and Partial Set Partitions, arXiv:2402.16292 [math.ST], 2024. See p. 21.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Apart from a(0), same as A006602, and after subtracting 1, A007411.

Cf. A000372, A003182, A006126, A014466, A261006, A304997, A304998, A306505, A306550, A321679.

First differences of A306505. - Gus Wiseman, Feb 23 2019

a(n) = A003182(n) - A003182(n-1) for n > 0. - Andrew Howroyd, May 28 2023

a(8)-a(9) added using A003182 by Andrew Howroyd, May 28 2023

A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

Original entry on oeis.org

2, 1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210, 789204635842035039135545297410259322

Offset: 0

Colin Mallows

Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000
Also number of antichain covers of an unlabeled n-set, so a(n) equals first differences of A003182. - Vladeta Jovovic, Goran Kilibarda, Aug 18 2000
Also number of inequivalent (under permutation of variables) nondegenerate monotone Boolean functions of n variables. We say h and g (functions of n variables) are equivalent if there exists a permutation p of S_n such that hp=g. E.g., a(3)=5 because xyz, xy+xz+yz, x+yz+xyz, xy+xz+xyz, x+y+z+xy+xz+yz+xyz are 5 inequivalent nondegenerate monotone Boolean functions that generate (by permutation of variables) the other 4. For example, y+xz+xyz can be obtained from x+yz+xyz by exchanging x and y. - Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 16 2006
The non-spanning/covering case is A003182. The labeled case is A006126. - Gus Wiseman, Feb 20 2019

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(4) = 20 antichains:
  {}    {{1}}  {{12}}    {{123}}         {{1234}}
  {{}}         {{1}{2}}  {{1}{23}}       {{1}{234}}
                         {{13}{23}}      {{12}{34}}
                         {{1}{2}{3}}     {{14}{234}}
                         {{12}{13}{23}}  {{1}{2}{34}}
                                         {{134}{234}}
                                         {{1}{24}{34}}
                                         {{1}{2}{3}{4}}
                                         {{13}{24}{34}}
                                         {{14}{24}{34}}
                                         {{13}{14}{234}}
                                         {{12}{134}{234}}
                                         {{1}{23}{24}{34}}
                                         {{124}{134}{234}}
                                         {{12}{13}{24}{34}}
                                         {{14}{23}{24}{34}}
                                         {{12}{13}{14}{234}}
                                         {{123}{124}{134}{234}}
                                         {{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{34}}
(End)

Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, Discrete Multivariate Analysis. MIT Press, 1975, p. 34. [In part (e), the Hierarchy Principle for log-linear models is defined. It essentially says that if a higher-order parameter term is included in the log-linear model, then all the lower-order parameter terms should also be included. - Petros Hadjicostas, Apr 10 2020]
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
A. A. Mcintosh, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 14.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11(4) (1999), 127-138 (translated in Discrete Mathematics and Applications, 9(6) (1999), 593-605).
C. Lienkaemper, When do neural codes come from convex or good covers?, 2015.
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126 (labeled case), A007411, A014466, A261005, A293993, A304997, A304998, A304999, A305001, A305855, A306505, A320449, A321679.

a(n) = A007411(n) + 1.

First differences of A003182. - Gus Wiseman, Feb 23 2019

a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31 2000
Entry revised by N. J. A. Sloane, Jul 23 2006
a(7) from A007411 and A003182. - N. J. A. Sloane, Aug 13 2015
Named edited by Petros Hadjicostas, Apr 08 2020
a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022
a(9) from A007411. - Dmitry I. Ignatov, Nov 27 2023

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 8, 72, 1824, 220608, 498243968, 309072306743552, 14369391925598802012151296, 146629927766168786239127150948525247729660416

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

			The a(2) = 8 antichains:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 12.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A003182, A014466, A293606, A293993, A306505, A320449, A321679.
Cf. A245079.

Binomial transform of A304999.

Inverse binomial transform of A245079. - Aniruddha Biswas, May 11 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 11 2024

A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 5, 87, 6398, 7745253, 2414573042063, 56130437190053518791691, 286386577668298410118121281898931424413687

Offset: 0

Gus Wiseman, May 23 2018

nonn

From Gus Wiseman, Jul 03 2019: (Start)
Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
(End)

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

The binomial transform is the non-covering case A307249.
The second binomial transform is A014466.
Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

a(9) from A307249 - Dmitry I. Ignatov, Nov 27 2023

A305857 Number of unlabeled intersecting antichains on up to n vertices.

Original entry on oeis.org

1, 2, 3, 6, 15, 87, 3528, 47174113

Offset: 0

Gus Wiseman, Jun 11 2018

An intersecting antichain S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection, and none of which is a subset of any other.

			Non-isomorphic representatives of the a(4) = 15 intersecting antichains:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,4},{2,4},{3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Cf. A001206, A006126, A051185, A261006, A283877, A304998, A305843, A305844, A305854, A305855, A305856.

a(n) = A305855(0) + A305855(1) + ... + A305855(n). - Brendan McKay, May 11 2020

a(6) from Andrew Howroyd, Aug 13 2019

a(7) from Brendan McKay, May 11 2020

A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 18, 142, 3100, 823042

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 18 antichains:
{{1,2,3}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

A304996 Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 6, 24, 166, 3266, 826308

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 24 antichains:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{1},{2}}
{{2},{1,2}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{1},{2},{1,2}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Partial sums of A304997.
Cf. A006126, A261005, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A014466, A293993, A319639, A319721, A326704, A326972.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A317073 Number of antichains of multisets with multiset-join a normal multiset of size n.

Original entry on oeis.org

1, 1, 3, 16, 198, 9890, 8592538

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is normal if it spans an initial interval of positive integers. The multiset-join of a set of multisets has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 16 antichains of multisets:
  (111),
  (122), (12)(22), (1)(22),
  (112), (11)(12), (2)(11),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A255906, A285572, A303837, A303838, A304998, A305001.

Cf. A317074, A317075, A317077, A317079.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
Table[Length[Join@@Table[auu[m],{m,allnorm[n]}]],{n,5}]

a(6) from Robert Price, Jun 21 2021

cp's OEIS Frontend

A002494 Number of n-node graphs without isolated nodes.

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A305854 Number of unlabeled spanning intersecting set-systems on n vertices.

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A261005 Number of unlabeled simplicial complexes with n nodes.

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A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

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A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

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A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

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A305857 Number of unlabeled intersecting antichains on up to n vertices.

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A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

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